What is the name of the line around the circle? The circle has a line around it, does it have a name?
The line is called the circle. The domain inside the line is correctly (scientifically) called a disc (UK)/disk (US). From Wikipedia:
Circumference: the length of one circuit along the circle, or the distance around the circle.
Disc: the region of the plane bounded by a circle.
In the introduction of the Wikipedia entry:
In everyday use, the term "circle" may be used interchangeably to refer to either the boundary of the figure, or to the whole figure including its interior; in strict technical usage, the circle is only the boundary and the whole figure is called a disc.
Perimeter seems to mean both the line and its length:
A perimeter is a path that surrounds a two-dimensional shape. The term may be used either for the path or its length—it can be thought of as the length of the outline of a shape. The perimeter of a circle or ellipse is called its circumference.
The term perimeter refers either to the curve constituting the boundary of a lamina or else to the length of this boundary.
The perimeter of a circle is called the circumference, although that term is used by some authors to refer to the perimeter of an arbitrary curved geometric figure.
Still using mathematical terminology, if you want to, you can call the line the boundary of the circle (or boundary of the disc), and even hairsplitting mathematicians will find that correct.